Dislocations edge dislocation elastic fields

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

We have now succeeded in identifying the deformations induced by a screw dislocation, and have also found a generic formula for the fields due to an arbitrary dislocation. Our present task is to return to the edge dislocation for the purposes of completeness and because this solution will make its way into our future analysis. This is particularly evident in the case of isotropic linear elasticity where we will find that even for dislocations of mixed character, their geometries may be thought of as a superposition of pure edge and pure screw dislocations. 

Earlier we made the promise that the energy stored in the elastic fields had a generic logarithmic character, regardless of the type of the dislocation. To compute the elastic strain energy associated with the edge dislocation we use eqn (8.16). In the context of the stress fields given above,... 

Edge dislocation fields using the elastic Green function... 

The displacement field of an edge-dislocation in the SmA phase has been calculated by De Gennes in the framework of the elastic continuum theory... 

Figure 10.7. Idealized model of an edge dislocation for the calculation of the elastic energy the displacement field, which spreads over many atomic sites in the x and y directions around the core (shaded area on left panel) is assumed to be confined on the glide plane (xz), as shown schematically on the right panel. The infinitesimal dislocation at x gives rise to a shear stress axy(x, 0) at another point x, where the displacement is (x).

Integrating this expression over all values of x from —LtoL (with L large enough to accommodate the range of the displacement field), and over dbg(x ) to account for the contributions from all infinitesimal dislocations, we obtain for the elastic energy of the edge dislocation... 

The superscripts e and 5 refer to edge and screw and serve as an instruction to use the isotropic linear elastic stress fields for the edge and screw dislocation, respectively, but with the Burgers vector adjusted to account for relevant trigonometric weighting factors. 

---